So far we have constructed a phenomenological single-nucleon potential \(U_J(\rho ,\delta ,\mathbf {k})\)  in the form of (REF ). The resulting EOS of ANM can safely fulfills the empirical constraints (a)-(f) mentioned at the beginning of this subsection.
In particular, the EOS of SNM is consistent with the collective flow constraint on the pressure [1]} and the optical potential constraint on the momentum dependence of the isoscalar potential in SNM [2]}. In addition, the EOS of PNM from our phenomenological potential is consistent with the microscopic calculations using chiral effective interactions [3]}. Moreover, the symmetry energy around \(\rho _0\)  is also consistent with existing constraints from analyzing various terrestrial experiments (e.g, the analysis of isobaric analog states [4]}) and neutron star observations.
Besides, the potential (REF ) thus designed is also consistent with the HVH theorem, i.e., the isovector symmetry potential and the second-order symmetry potential should be related as in Eq. (REF ).
Of course, here we have no desire to construct a celebrated effective model for the EOS of ANM. Indeed, advanced nuclear many-body theories either phenomenological or microscopic can predict numerically more accurately the EOS of neutron-rich matter at certain densities and/or isospin asymmetries. However, analytical expressions for the \(U_J(\rho ,\delta ,\mathbf {k})\)  and all the characteristics of the corresponding EOS with only 9 parameters that can describe satisfactorily essentially all known constraints on the EOS are invaluable for many purposes. As we shall show next, serving as a reliable basis in the \(\epsilon \) -expansion approach these expressions facilitate the study of nuclear EOSs in spaces with dimensions different from 3.
